7 Main Theorem

Theorem 35 Approximate Gumbel Convergence: Critical Threshold at \(\alpha = 2/3\)

Assume the properties in Definitions 27, 28, and 29. Let \(N_n = \lfloor n^\alpha \rfloor \) for some \(\alpha {\gt} 0\). For a sequence of Gumbel grids \(Y^{(n)}\):

(Convergence for \(\alpha {\gt} 2/3\)) If \(\alpha {\gt} 2/3\), then for any \(r \in \mathbb {R}\):

\[ \mathbb {P}\left(\frac{T_{\text{Approx}}^{N_n}(n) - C_g \cdot n}{\sigma _g \cdot n^{1/3}} \leq r\right) \to F_{\text{GUE}}(r) \]

as \(n \to \infty \). That is, the approximate Gumbel LPP converges to the same GUE Tracy-Widom distribution as the exact Gumbel LPP.
(Divergence for \(\alpha {\lt} 2/3\)) If \(\alpha {\lt} 2/3\), then the fluctuations diverge:

\[ \frac{T_{\text{Approx}}^{N_n}(n) - C_g \cdot n}{\sigma _g \cdot n^{1/3}} \xrightarrow {\mathbb {P}} +\infty \]

as \(n \to \infty \). More precisely, for any \(M {\gt} 0\):

\[ \mathbb {P}\left(\frac{T_{\text{Approx}}^{N_n}(n) - C_g \cdot n}{\sigma _g \cdot n^{1/3}} {\gt} M\right) \to 1 \]

Thus \(\alpha = 2/3\) represents a sharp threshold: the approximation parameter \(N\) must grow faster than \(n^{2/3}\) for the limiting distribution to remain Tracy-Widom GUE.

Proof ▶

By Theorems 23 and 24:

\[ 2n \cdot h_{N_n}\left(\frac{T_{\text{Gumbel}}(n)}{2n}\right) \leq T_{\text{Approx}}^{N_n}(n) - T_{\text{Gumbel}}(n) \leq \frac{1}{N_n} L_{\text{Exp}}(n) \]

Dividing by \(\sigma _g n^{1/3}\):

\[ \frac{2n \cdot h_{N_n}(T_{\text{Gumbel}}(n)/(2n))}{\sigma _g n^{1/3}} \leq \frac{T_{\text{Approx}}^{N_n}(n) - T_{\text{Gumbel}}(n)}{\sigma _g n^{1/3}} \leq \frac{L_{\text{Exp}}(n)}{N_n \sigma _g n^{1/3}} \]

Case 1: \(\alpha {\gt} 2/3\).

By Definition 29, \(L_{\text{Exp}}(n)/n \xrightarrow {\mathbb {P}} D_L\). We have:

\[ \frac{L_{\text{Exp}}(n)}{N_n \sigma _g n^{1/3}} = \frac{L_{\text{Exp}}(n)}{n} \cdot \frac{n}{N_n \sigma _g n^{1/3}} \]

By Lemma 34, \(n/(N_n n^{1/3}) \to 0\) when \(\alpha {\gt} 2/3\). Therefore by Lemma 33:

\[ \frac{T_{\text{Approx}}^{N_n}(n) - T_{\text{Gumbel}}(n)}{\sigma _g n^{1/3}} \xrightarrow {\mathbb {P}} 0 \]

By Definition 27:

\[ \frac{T_{\text{Gumbel}}(n) - C_g n}{\sigma _g n^{1/3}} \xrightarrow {d} F_{\text{GUE}} \]

Applying Slutsky’s Theorem (Theorem 32) gives:

\[ \frac{T_{\text{Approx}}^{N_n}(n) - C_g n}{\sigma _g n^{1/3}} \xrightarrow {d} F_{\text{GUE}} \]

Case 2: \(\alpha {\lt} 2/3\).

By the lower bound from Theorem 24 and Lemma 17, for \(x {\gt} 0\):

\[ h_{N_n}(x) \geq \frac{e^{-x}}{3N_n} \]

By Definition 28, \(T_{\text{Gumbel}}(n)/n \xrightarrow {\mathbb {P}} D_\ell \) where \(D_\ell {\gt} 0\). Therefore \(T_{\text{Gumbel}}(n)/(2n) \xrightarrow {\mathbb {P}} D_\ell /2 {\gt} 0\), which means for large \(n\), \(T_{\text{Gumbel}}(n)/(2n)\) is bounded away from zero with high probability. Thus:

\[ \frac{2n \cdot h_{N_n}(T_{\text{Gumbel}}(n)/(2n))}{\sigma _g n^{1/3}} \geq \frac{2n \cdot e^{-T_{\text{Gumbel}}(n)/(2n)}}{3N_n \sigma _g n^{1/3}} \geq \frac{C \cdot n}{N_n n^{1/3}} = C \cdot n^{2/3 - \alpha } \]

for some constant \(C {\gt} 0\) (with high probability). When \(\alpha {\lt} 2/3\), the exponent \(2/3 - \alpha {\gt} 0\), so this lower bound diverges to \(+\infty \) as \(n \to \infty \). Therefore:

\[ \frac{T_{\text{Approx}}^{N_n}(n) - T_{\text{Gumbel}}(n)}{\sigma _g n^{1/3}} \xrightarrow {\mathbb {P}} +\infty \]

Since the scaled \(T_{\text{Gumbel}}(n)\) converges in distribution (hence is tight), the scaled \(T_{\text{Approx}}^{N_n}(n)\) must diverge to \(+\infty \) in probability.